Problem 28 - Algebra A Combined Function

Question

If \(f(x)=\frac{x+8}{2 x-1}\) and \(g(x)=\frac{x-2}{x-5},\) find each function value. $$ g(-5) $$

Step-by-Step Solution

Answer

The value of \( g(-5) \) is \( \frac{7}{10} \).

Step 1: Understand the function

The given function is \( g(x) = \frac{x-2}{x-5} \). To solve for \( g(-5) \), we need to substitute \( x = -5 \) into this function.

Step 2: Substitute the value into the function

Substitute \( x = -5 \) into \( g(x) = \frac{x-2}{x-5} \). This gives us \( g(-5) = \frac{-5-2}{-5-5} \).

Step 3: Perform the calculations

Calculate the numerator: \( -5 - 2 = -7 \). Next, calculate the denominator: \( -5 - 5 = -10 \). So, \( g(-5) = \frac{-7}{-10} \).

Step 4: Simplify the expression

The simplified form of \( \frac{-7}{-10} \) is \( \frac{7}{10} \) because a negative divided by a negative is positive.

Key Concepts

Function Evaluation

Function evaluation is the process of finding the value of a function at a specific point. To evaluate the function, you simply substitute the given value into the variable of the function and perform the necessary operations. For instance, if we have a function \( g(x) = \frac{x-2}{x-5} \), and we want to find \( g(-5) \), we substitute \( x = -5 \) into the function. This substitution results in \( g(-5) = \frac{-5-2}{-5-5} \). After plugging in the value, it becomes a straightforward arithmetic problem. By solving this, you effectively learn what output the function produces when the input is \( -5 \). Remember, the key part of function evaluation is accurately substituting the given value into every occurrence of the variable.

Numerator and Denominator

In the context of rational functions, understanding the numerator and the denominator is crucial. These are the two components of any fraction where the numerator is the top part and the denominator is the bottom part. For the function \( g(x) = \frac{x-2}{x-5} \), the numerator is \( x - 2 \), and the denominator is \( x - 5 \). When evaluating a function like \( g(-5) \), you calculate each part separately:

• Numerator: Substitute \( x = -5 \), compute \( -5 - 2 = -7 \).
• Denominator: Again substitute \( x = -5 \), compute \( -5 - 5 = -10 \).

By understanding each part, it can help in error-checking calculations and ensures that you correctly apply operations. Without both parts, the rational function wouldn’t be complete.

Simplifying Fractions

After evaluating a function and obtaining a fraction, simplifying fractions is the next step. This process involves reducing the fraction to its smallest form. In our example, we obtained \( g(-5) = \frac{-7}{-10} \). Simplification checks if the result can be presented in a more readable form. Here, notice that both the numerator and the denominator are negative. When both signs are negative, they cancel each other out, making the fraction positive. Thus, \( \frac{-7}{-10} \) simplifies to \( \frac{7}{10} \).

• Recognize opposite signs in both parts and use the rule that a negative divided by a negative equals a positive.
• Check for common factors (none in this case) to reduce further if possible.

Understanding fraction simplification is essential to providing clean, final answers that are easier to interpret and use in further calculations.